Use a point cutting roundover bit, with a template and guide bushing.
Melting should not be an issue at all when cutting roundovers.
@ger21 described exactly how I would carry out the project with one exception, and that is the choice of router bit. I would also heed the advice of
@gdonham1 and use a high feed rate in order to avoid melting. My choice of router bit is based on the width of the already existing grooves of which
@Martyc wants to round the edges. A point cutting roundover bit would not apply a full 90° roundover in a single pass while an ovolo, a.k.a plunge roundover with a 1/4 inch small diameter would.
While I suggested a Freud ovolo bit, I could not find a Freud point cutting roundover bit!
The two .pdf files linked are exactly the same images as the .png which show the images directly.